Potpourri: Links and a French Curve Question

Links
Here's the one I most want to share with everyone, but now I have some reservations...
A lovely documentary on mathematical origami, called Between the Folds, has been posted to youtube. The DVD costs $20, which seems quite reasonable to me. This youtube posting is not from the producers, Green Fuse Films. I've contacted them, in case they want to have it removed. Watch it quickly if you'd like to. The official trailer is also on youtube, so you can get a taste, even after the pirate version is gone.



Many of the links I'm saving these days are ideas I hope to share with my students:

Calculus I
Intuiting the Chain Rule (Girls' Angle)

Calculus II
Volume of a Pendant (SquareCircleZ)

Linear Algebra
What's so special about the 4 fundamental subspaces? (math.stackexchange)



Some of the links aren't for a course, but for me to try out, some day when I have time...

Geometry
Circles and Equilateral Triangles (Pat's Blog)

Question
A student came to me last week for help. He wants to use fabric (a very loose weave, which will stretch some) to cover a sphere. He needed to know what shape to cut. I had no idea, and first suggested he look at information on world maps. He wants nice seams and no puckering. I thought a shape that wrapped around the equator, with sort of triangular tabs up and down to the poles might work. We knew the sides of those needed to curve, and he wanted an equation. I thought he should experiment with 8 tabs in each direction at first, and those would need to have a 45 degree angle at the pole. I had no idea what kind of equation would fit this.

A former student, who designs and sews clothing, was in the math lab, and I brought him into our conversation. He said he would use a French curve, which is what you see in the image above. We looked it up, and no one seems to know the equations for the curves.

So my question is about the French curves. Can anyone help me figure out their equations?

The coolest thing happened as I began to write this post. I was trying to sketch my pattern idea freehand, and I'm a terrible drawer. So I turned to Geogebra, and started putting in the points and lines. When I got to the curved segments, I was in the right frame of mind. I knew I needed something between the points (0,.5) and (1,3.5), with a vertical tangent at (0,.5) and a slope of 1 (same as 45 degrees) at (1,3.5). I knew that the square root function starts out with a vertical tangent, and so I figured I'd try to modify that. Getting it to go though those two points, I was having trouble getting the slope at the top right. (I think there's a way...) So I figured I could change what root I used. And I solved the problem I had posed! (Always a rush.) The first curve is y = 3 times the cube root of x. The next one in the same direction is y = 3 times the cube root of (x-2), and the one between, that goes in the other direction, is y = 3 times the cube root of (2-x). They may not be just the right shape. My student will have to experiment at this point, I think.

Or is there a way to figure out a better shape mathematically? The problem is that we are really just approximating, because the fabric is a flat surface, and the sphere is everywhere curved (in all directions). So I don't see how we can
 describe perfectly what we want. Ideas?

[Oops! I just tried to explain this to my son, and realized those points have two 45 degree angles in them, for a total of 90 degrees. Back to the drawing board! (More to come...)]

Calculus: Solids of Revolution

In about a week, I'll be working with my students on solids of revolution. (At least, I hope to be. My calc II class is too small right now, and could be canceled.) Patrick Honner's Math Photo post of three beautiful bottles exemplifying solids of revolutions inspired me. I looked up beautiful bottles on Google Images, and there are so many. Some of them are solids of revolution and some aren't. I wonder if my students would benefit by identifying which are which.

Maybe that could be a first step, and then drawing a curve they think would make a good-looking volume when revolved. Hmm... (Anyone know a super-easy 3D modeler we good put our curves in, and get visuals from?)

Maybe this Friday I can get some students to come in and work with me on making models of volumes (not volumes of revolution, though), like these that Bowman Dickson made, or these (both types) that Rebecka Peterson made.

Wolfram Alpha Glitch!

There are many great math calculation resources online. Wolfram Alpha is probably the most well-known. I've used it often from home, where I don't have a calculator. I used to create graphs there that I could copy onto my handouts and tests, but now I use Desmos for that.

Wolfram has lots of interesting features, but it must be hard to make sure everything works just right. There is a statistical procedure to find the line or curve of best fit given a number of data points. If the data points are exactly on a line or curve, you'd think it would be especially easy. But somehow it isn't.

I was asking Wolfram to give me an equation like y=ax²+bx+c, which would go as near as possible to the points (0,0), (1,1), (2,4). The right answer is y=x², but Wolfram gives an answer with teeny tiny x and constant terms added in.

A quick Google search doesn't turn up any information on why this happens. I learned about it from David Cushing on Aperiodical. It turns out that if you ask for a parabola instead of a quadratic, you get the right answer. The wrong answer happens because the computer is using a complex procedure which produces numbers that can only be represented approximately on the computer, and some very small round-off errors show up at the end.

Explaining Calculus using the Up-Goer Five Text Editor

Randall Munroe writes xkcd, a series of comic strips that often involve math, technology, and science. (Sometimes not safe for work, or kids.) A few years back, he wrote a description of the Saturn Five, which he called the Up-Goer Five, using only the one thousand most commonly used words. Theo Sanderson was intrigued, and built a text editor that only allows those words. He called it 'The Up-Goer Five Text Editor.'

A friend linked to this on Facebook, asking: "Can you explain a hard idea using the ten hundred most common words?" So I decided to try to explain calculus. What I wrote seems pretty bad. My explanation from October seems much clearer. But I thought I'd share.

Will you try it with a math topic? (I'd love to see a better description of calculus.) Here are some of the words you can't use: math, slope, steepness, circle, curve, and infinity.


About calculus:Some lines are straight. Some are not. It's easy to think about how much a straight line goes up as it goes over. It's harder to think about that when the line is not straight. Two guys figured out some good ideas to help think about this around 1670. They cared because it would help people think about the world. How fast do things fall? How does light turn when it goes through round glass? How do we find the best way to do something? Their ideas also help us think about areas that don't have straight edges.

To think about these things, we need to think about something bigger than any number, about  two things closer than any number can say, and about cutting smaller parts than any number can say. Thinking about this in just right ways takes some hard thinking.

Math Monday Blog Hop: Multiplication and Division

Cindy, over at love2learn2day, runs a weekly blog hop on elementary level math. This week's topic is multiplication and division. I'm writing this post to gather a few resources I like.

Most of the posts that usually gather in the blog hop are activities. Before we get to activities, we might want to ask ourselves some questions, like "What is multiplication?" Multiplication can be modeled in many ways, and it's important to give young kids a chance to think about it through lots of different frameworks. Your own conceptions of multiplication may grow as you think about it.

One resource that will help you broaden your perspective on multiplication (along with a number of other math topics) is the lovely book Moebius Noodles. (It's $15 for the paperback version, and whatever you decide to pay for the pdf version.) It is full of activities to share with young kids.

I also love the art connection made by Waldorf schools, with their multiplication stars. Beauty helps us love what we're doing, which helps us learn.


As we teach, the more we can see the child's understanding, and help that grow, instead of telling them what is so, the better off we are. I can think of two ways we might just tell children our adult understanding, without giving them enough opportunities for exploration. One is in the commutative nature of multiplication, noticing that 3x5 = 5x3. Kids don't see that at first. If we tell them, we take away their chance to discover it. I never knew that until I read ‘Third Graders Explore Multiplication’, a chapter by Virginia Brown in the book What's Happening In Math Class, Volume 1, edited by Deborah Schifter. [You might be interested to know that mathematicians study versions of multiplication that are not commutative. Matrices are a tool for solving systems of equations, if A and B are matrices, AB may not equal BA.]

The other example I'd like to share of this is our knowledge that to multiply by ten we just "add a zero". (Why does that work?) Here's an excerpt from the chapter 'Trust, Montessori Style', by Pilar Bewley in the soon-to-be-published book Playing With Math (edited by me):

Answer Versus Process 
Five-and-a-half-year-old Roland came to ask if he could multiply 8,696 times 10 using the stamp game. I was thrilled to see his interest in math taking off again after an unfortunate temper tantrum with the addition blank chart. I suggested he borrow a stamp game box from another classroom to supplement the one we have in ours, and he got started.

It took him a while to figure out where he would do the work, and then he painstakingly began to make 8,696 with the stamps… Six units, nine tens, six hundreds, and eight thousands… Leave a space and repeat… Six units, nine tens, six hundreds, and eight thousands…

He made the amount five times before it was time to go home, and he left his work out so he could return to it the next day. (Can you imagine what that looks like, once you’ve made 8,696 with little colored tiles ten times? How cool!)

“I’m going to get right back to work as soon as I change my shoes,” he declared before leaving that afternoon. “I won’t even talk to anybody!”

I waited eagerly for him to arrive the next morning, looking forward to the moment when he would put AAAAAALLLLLL those tiles together in neat rows by category, and he would have to exchange several times (not to mention his surprise at seeing all the units disappear when multiplying by ten).

Instead Roland came in, shook my hand, and said: “My dad told me that all I have to do is add a zero to 8,696 and I’ll have my answer, because when you multiply by ten you just add a zero.”

My heart sank. Oh no, Dad! You robbed your son of such a cool experience! He was getting ready to see what ten times 8,696 looks like, and would have discovered the process that takes place during multiplication. He won’t be doing multiplication tables for at least two more years in public school - the answer doesn’t matter yet, but the process could have really taught him something valuable.

Although I firmly believe that math is about understanding, multiplication facts are one of the few things that need to be memorized. Many students come to college not knowing those basic facts. When I was teaching beginning algebra, I gave a quick quiz on all the basic multiplication facts, and required students to keep trying until they could get 85% right. Here are some of the suggestions I offered them:

• Try some of the games at Math Playground.
• Use these flash cards, which help you visualize the numbers. 
• Play the Product Game.

Those suggestions were for adults who were likely to have emotional baggage about not yet knowing something basic. For kids, just have fun. Make up games (like John Golden does) using dice or cards, go on scavenger hunts, or tell stories. (Perhaps you could tell stories about the large family, who have to buy in bulk because they have so many children. Each child needs 3 pair of shoes, for dress-up, play, and beach. How many shoes are in that house?)

Hmm, all of that and I haven't even mentioned division. I guess that will have to wait for its own post.

More Links: Good article, video, tool, problems, game, and activity

What I've stumbled upon in the past two days:

• Can't keep up with the good articles I'd like to read. The De Morgan Forum pointed me to an article by Liping Ma, whose writing always gets me thinking. 
• Malke's love for the math found in stars primed me to take notice of this video by Toomai on stellations. He notices something intriguing that I'd like to explore when I have more time...
•  I think some day I'll want to explore the tools available at SageMath.

And some older ones (as I begin to slowly clear out my backlog...):

• Megan's thoughts on ranking tasks as a way to get students to grapple with definitions. Hmm, I wonder if I could design a good activity for my calculus or linear algebra courses.
• The link takes you to an illustration of adding cubes, but the page that's on is a huge collection of good problems.
• Kate's logic game, whose name has not been decided, looks great. I'd use it in Discrete Math. Reminds me of an old game called WFF 'n Proof.
• On inventing number systems (with third graders).

Links versus Real Writing

I used to share links more often. I used to write substantive blog posts more often, too. Since I've been writing less, I haven't been comfortable sharing lots of links. Didn't want this blog to descend into just a link-share. But it would be helpful to me to have them here. So maybe I'll start sharing my almost daily finds, even if it's not exciting for y'all.

• What is a radian? Here's Sam's colleague's applet (on geogebratube).
• Trig war for review (thanks again, Sam!)
• Measuring the moon's size through geometry (Anna Weltman)
• With an Eye on the Mathematical Horizon, by Deborah Ball (good article, can only be read online)
• Fermat's Little Theorem (the ad before this video is bizarre - if the ad stays the same...)

These were the tabs I've kept open - some for weeks - hoping to figure out how to use some, how to find time to really process others.

Online Conversations: Math Communication, and Understanding Computer Graphing

I am enjoying two online conversations right now.

Michael Pershan asked:

Students don't like to write about their reasoning. They don't present their work in a way that allows anyone else to comprehend their path to a solution. But we want kids to write about their reasoning. Conflict! Drama!

Why do kids hate writing about math?

I am currently trying to grade my students arguments (as prosecutors) for the murder mystery. Some of them really got into it. Most still didn't explain the math well. My take on this is that students will write (maybe even well) if we give them a good enough context.


In the other conversation, Mr. Honner blogged about what happens when you zoom in super far on Desmos, looking for the hole in a rational function. It gets a bit crazy. The conversation got more interesting for me when Alan Eliasen started explaining "interval arithmetic", which I had never heard of.

Starting Circle Trig in Pre-Calc

I'm teaching four classes this semester, which is a lot for me. That's embarrassing to admit - I know most math bloggers are high school teachers, and teach way more hours a week than I do, with more responsibilities for their students. But for me it's a heavy load. So I'm not prepping as much as usual. I've taught calc and pre-calc dozens of times, so I can usually get by with winging it. And, once in a while, I'm able to conduct a better class by improvising than I ever could have with a tight plan.

That's what happened yesterday in pre-calc. The day before that I had worked hard to get their tests graded, so in the morning I printed out the new unit sheet, and walked into class not particularly sure how I wanted to get us started. I had grabbed a problem from my computer, and asked them to start thinking about it while I handed back tests.


The problem:

Consider three circles, all tangent (externally). Their radii are 4 in, 5in, and 6in. What is the area between them? 

I had asked the students to draw a picture. After they had had plenty of time, I drew my picture on the board. Then I asked them how we might start thinking about the problem. A student suggested finding the area of the triangle formed by connecting the centers. I asked if that triangle's sides actually went through the points of tangency. No one answered. Unlike in a math circle, I rescued them be showing a picture of one circle with a tangent line, and reminding them that they likely proved in geometry that the tangent is perpendicular to the radius (the one that ends at the point of tangency). I don't know what that proof would look like. To me, it seems obvious because of the symmetry. (In the afternoon class, they didn't think it needed proving. It already looked necessary to them.)

To find the area of the triangle, one student suggested drawing in the height. We drew it in, but couldn't yet see how to find its length. One of the students suggested that we could find the measures of the angles. They first suggested using law of sines. That didn't work, so we used law of cosines. Sine of that angle gave the height over a triangle side, so we got the height, which gives us area of the triangle. Then we got the other angles and found the sector areas. The afternoon class did it without the height, so they got to use law of sines.

It was a lot of steps for them, but it was a great review of the triangle trig we'd done earlier in the semester. And maybe they got a small taste of what problem-solving looks like.

When we were done, I had just enough time to explain radians to the morning class. The afternoon class had more time, so we worked out the new circle-based definitions of the trig functions.

What are our intuitions about temperature?

I'm teaching exponential functions and logarithms in pre-calc right now. That means it's time to pull out my murder mystery, in which they will use logarithms to solve an important problem - which of their classmates killed John Doe? Since the murder mystery uses temperature to find the killer, I want to lead in with some thinking about how temperature changes over time.

On Wednesday and Thursday, I told my classes a story, and asked them to draw a graph. I said I was mixing some cake batter up to make a Halloween cake. I asked what temperature it should cook at. We decided to set our oven at 350 degrees. (In one class, I talked about how silly the Fahrenheit temperature scale is, but how, even with Centigrade, zero is just attached to water freezing. It's not the same as zero length, volume, or weight. Temperature is different...)

I also asked what temperature the batter was now. They told me room temperature, and we decided that was about 70 degrees. Then I drew axes on the board, labelled them, and asked the students to graph the temperature of the batter over time. Only one person (out of over 50 in the two classes) came close to the right shape. No one seems to have much intuition about how temperature changes. I did this once before, with the cooling coffee we always think about, and got slightly better results.

Here are my approximations of what students thought:

The green one may have been influenced by our attention in the past week to exponential growth, while the purple one seems to have taken the exponential growth we were studying and limited it by the temperature of the oven. I have often seen students give a linear graph like the blue one, and a logistic-like graph like the orange one. No one wants stuff to heat up fast at first, and then slower.

What makes their intuition bad here? Is there a physical experiment / demonstration we could do to improve their intuition? What would make exponential decay feel like the natural choice to them? Maybe cake is the wrong object to be heating?

Please help me think about this.